Graph Homomorphisms for Quantum Players

A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y. This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Most importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity.Published versio

Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number

We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. We also prove an analog of Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.Comment: 18 page

Curious properties of free hypergraph C*-algebras

A finite hypergraph $H$ consists of a finite set of vertices $V(H)$ and a collection of subsets $E(H) \subseteq 2^{V(H)}$ which we consider as partition of unity relations between projection operators. These partition of unity relations freely generate a universal C*-algebra, which we call the "free hypergraph C*-algebra" $C^*(H)$. General free hypergraph C*-algebras were first studied in the context of quantum contextuality. As special cases, the class of free hypergraph C*-algebras comprises quantum permutation groups, maximal group C*-algebras of graph products of finite cyclic groups, and the C*-algebras associated to quantum graph homomorphism, isomorphism, and colouring. Here, we conduct the first systematic study of aspects of free hypergraph C*-algebras. We show that they coincide with the class of finite colimits of finite-dimensional commutative C*-algebras, and also with the class of C*-algebras associated to synchronous nonlocal games. We had previously shown that it is undecidable to determine whether $C^*(H)$ is nonzero for given $H$. We now show that it is also undecidable to determine whether a given $C^*(H)$ is residually finite-dimensional, and similarly whether it only has infinite-dimensional representations, and whether it has a tracial state. It follows that for each one of these properties, there is $H$ such that the question whether $C^*(H)$ has this property is independent of the ZFC axioms, assuming that these are consistent. We clarify some of the subtleties associated with such independence results in an appendix.Comment: 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph C*-algebra', added Remark 2.2

Graph-theoretical Bounds on the Entangled Value of Non-local Games

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2014